This is what I had in mind:

#!/usr/bin/perl
use warnings;
use strict;

use Test::More tests => 2 * 2;

use Sub::Override;

my @warnings;
my $override;

use Logger;
BEGIN {
    $override = 'Sub::Override'->new(
        'Logger::log_warn' => sub { push @warnings, $_[0] }
    );
}
use MyObj;

for my $test ([1, undef, 'no warnings'], [11, 'Value too large', 'warn
+ings']) {
    my ($value, $warnings, $name) = @$test;
    @warnings = ();
    my $o = bless {value => $value}, 'MyObj';
    ok($o, 'constructs');
    $o->foo;
    is($warnings[0], $warnings, $name);
}
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but I hadn't tested it. It does in fact fail, but only because you have a bug that needs fixed in the code being tested :-)

Edit: Turns out when you replace the Logger sub, Moose::Exporter can't determine the sub name from the coderef. Export it like this and it works:

my ($import, $unimport, $init_meta) = Moose::Exporter->build_import_me
+thods(
    as_is => ['Logger::log_warn']
);
sub import { goto &$import }
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